Simplify the following expression and state the condition under which the simplification is valid. You can assume that $x \neq 0$. $a = \dfrac{x}{36x + 63} \div \dfrac{6x}{28x + 49} $
Explanation: Dividing by an expression is the same as multiplying by its inverse. $a = \dfrac{x}{36x + 63} \times \dfrac{28x + 49}{6x} $ When multiplying fractions, we multiply the numerators and the denominators. $a = \dfrac{ x \times (28x + 49) } { (36x + 63) \times 6x } $ $ a = \dfrac {x \times 7(4x + 7)} {6x \times 9(4x + 7)} $ $ a = \dfrac{7x(4x + 7)}{54x(4x + 7)} $ We can cancel the $4x + 7$ so long as $4x + 7 \neq 0$ Therefore $x \neq -\dfrac{7}{4}$ $a = \dfrac{7x \cancel{(4x + 7})}{54x \cancel{(4x + 7)}} = \dfrac{7x}{54x} = \dfrac{7}{54} $